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Theorem cpmadumatpolylem2 22786

Description: Lemma 2 for cpmadumatpoly 22787. (Contributed by AV, 20-Nov-2019.) (Revised by AV, 15-Dec-2019.)

**Hypotheses**
Ref	Expression
cpmadumatpoly.a	⊢ 𝐴 = (𝑁 Mat 𝑅)
cpmadumatpoly.b	⊢ 𝐵 = (Base‘𝐴)
cpmadumatpoly.p	⊢ 𝑃 = (Poly₁‘𝑅)
cpmadumatpoly.y	⊢ 𝑌 = (𝑁 Mat 𝑃)
cpmadumatpoly.t	⊢ 𝑇 = (𝑁 matToPolyMat 𝑅)
cpmadumatpoly.r	⊢ × = (._r‘𝑌)
cpmadumatpoly.m0	⊢ − = (-_g‘𝑌)
cpmadumatpoly.0	⊢ 0 = (0_g‘𝑌)
cpmadumatpoly.g	⊢ 𝐺 = (𝑛 ∈ ℕ₀ ↦ if(𝑛 = 0, ( 0 − ((𝑇‘𝑀) × (𝑇‘(𝑏‘0)))), if(𝑛 = (𝑠 + 1), (𝑇‘(𝑏‘𝑠)), if((𝑠 + 1) < 𝑛, 0 , ((𝑇‘(𝑏‘(𝑛 − 1))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑛))))))))
cpmadumatpoly.s	⊢ 𝑆 = (𝑁 ConstPolyMat 𝑅)
cpmadumatpoly.m1	⊢ · = ( ·_𝑠 ‘𝑌)
cpmadumatpoly.1	⊢ 1 = (1_r‘𝑌)
cpmadumatpoly.z	⊢ 𝑍 = (var₁‘𝑅)
cpmadumatpoly.d	⊢ 𝐷 = ((𝑍 · 1 ) − (𝑇‘𝑀))
cpmadumatpoly.j	⊢ 𝐽 = (𝑁 maAdju 𝑃)
cpmadumatpoly.w	⊢ 𝑊 = (Base‘𝑌)
cpmadumatpoly.q	⊢ 𝑄 = (Poly₁‘𝐴)
cpmadumatpoly.x	⊢ 𝑋 = (var₁‘𝐴)
cpmadumatpoly.m2	⊢ ∗ = ( ·_𝑠 ‘𝑄)
cpmadumatpoly.e	⊢ ↑ = (._g‘(mulGrp‘𝑄))
cpmadumatpoly.u	⊢ 𝑈 = (𝑁 cPolyMatToMat 𝑅)

**Assertion**
Ref	Expression
cpmadumatpolylem2	⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠))) → (𝑈 ∘ 𝐺) finSupp (0_g‘𝐴))

Distinct variable groups: 𝐵,𝑛 𝑛,𝑀 𝑛,𝑁 𝑅,𝑛 𝑆,𝑛 𝑛,𝑌 𝑛,𝑏 𝑛,𝑠 Allowed substitution hints: 𝐴(𝑛,𝑠,𝑏) 𝐵(𝑠,𝑏) 𝐷(𝑛,𝑠,𝑏) 𝑃(𝑛,𝑠,𝑏) 𝑄(𝑛,𝑠,𝑏) 𝑅(𝑠,𝑏) 𝑆(𝑠,𝑏) 𝑇(𝑛,𝑠,𝑏) · (𝑛,𝑠,𝑏) × (𝑛,𝑠,𝑏) 𝑈(𝑛,𝑠,𝑏) 1 (𝑛,𝑠,𝑏) ↑ (𝑛,𝑠,𝑏) 𝐺(𝑛,𝑠,𝑏) ∗ (𝑛,𝑠,𝑏) 𝐽(𝑛,𝑠,𝑏) 𝑀(𝑠,𝑏) − (𝑛,𝑠,𝑏) 𝑁(𝑠,𝑏) 𝑊(𝑛,𝑠,𝑏) 𝑋(𝑛,𝑠,𝑏) 𝑌(𝑠,𝑏) 0 (𝑛,𝑠,𝑏) 𝑍(𝑛,𝑠,𝑏)

**Proof of Theorem cpmadumatpolylem2**
Step	Hyp	Ref	Expression
1		fvexd 6841	. 2 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠))) → (0_g‘𝐴) ∈ V)
2		crngring 20149	. . . . . 6 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring)
3	2	anim2i 617	. . . . 5 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring))
4	3	3adant3 1132	. . . 4 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring))
5	4	ad2antrr 726	. . 3 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠))) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring))
6		cpmadumatpoly.s	. . . 4 ⊢ 𝑆 = (𝑁 ConstPolyMat 𝑅)
7		cpmadumatpoly.p	. . . 4 ⊢ 𝑃 = (Poly₁‘𝑅)
8		cpmadumatpoly.y	. . . 4 ⊢ 𝑌 = (𝑁 Mat 𝑃)
9	6, 7, 8	0elcpmat 22626	. . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (0_g‘𝑌) ∈ 𝑆)
10	5, 9	syl 17	. 2 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠))) → (0_g‘𝑌) ∈ 𝑆)
11		cpmadumatpoly.a	. . . . 5 ⊢ 𝐴 = (𝑁 Mat 𝑅)
12		cpmadumatpoly.b	. . . . 5 ⊢ 𝐵 = (Base‘𝐴)
13		cpmadumatpoly.r	. . . . 5 ⊢ × = (._r‘𝑌)
14		cpmadumatpoly.m0	. . . . 5 ⊢ − = (-_g‘𝑌)
15		cpmadumatpoly.0	. . . . 5 ⊢ 0 = (0_g‘𝑌)
16		cpmadumatpoly.t	. . . . 5 ⊢ 𝑇 = (𝑁 matToPolyMat 𝑅)
17		cpmadumatpoly.g	. . . . 5 ⊢ 𝐺 = (𝑛 ∈ ℕ₀ ↦ if(𝑛 = 0, ( 0 − ((𝑇‘𝑀) × (𝑇‘(𝑏‘0)))), if(𝑛 = (𝑠 + 1), (𝑇‘(𝑏‘𝑠)), if((𝑠 + 1) < 𝑛, 0 , ((𝑇‘(𝑏‘(𝑛 − 1))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑛))))))))
18	11, 12, 7, 8, 13, 14, 15, 16, 17, 6	chfacfisfcpmat 22759	. . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → 𝐺:ℕ₀⟶𝑆)
19	2, 18	syl3anl2 1415	. . 3 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → 𝐺:ℕ₀⟶𝑆)
20	19	anassrs 467	. 2 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠))) → 𝐺:ℕ₀⟶𝑆)
21		cpmadumatpoly.u	. . . 4 ⊢ 𝑈 = (𝑁 cPolyMatToMat 𝑅)
22	11, 12, 6, 21	cpm2mf 22656	. . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑈:𝑆⟶𝐵)
23	5, 22	syl 17	. 2 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠))) → 𝑈:𝑆⟶𝐵)
24		ssidd 3961	. 2 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠))) → 𝑆 ⊆ 𝑆)
25		nn0ex 12409	. . 3 ⊢ ℕ₀ ∈ V
26	25	a1i 11	. 2 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠))) → ℕ₀ ∈ V)
27	6	ovexi 7387	. . 3 ⊢ 𝑆 ∈ V
28	27	a1i 11	. 2 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠))) → 𝑆 ∈ V)
29	11, 12, 7, 8, 13, 14, 15, 16, 17	chfacffsupp 22760	. . 3 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → 𝐺 finSupp (0_g‘𝑌))
30	29	anassrs 467	. 2 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠))) → 𝐺 finSupp (0_g‘𝑌))
31		eqid 2729	. . . . . 6 ⊢ (0_g‘𝐴) = (0_g‘𝐴)
32		eqid 2729	. . . . . 6 ⊢ (0_g‘𝑌) = (0_g‘𝑌)
33	11, 21, 7, 8, 31, 32	m2cpminv0 22665	. . . . 5 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑈‘(0_g‘𝑌)) = (0_g‘𝐴))
34	2, 33	sylan2 593	. . . 4 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → (𝑈‘(0_g‘𝑌)) = (0_g‘𝐴))
35	34	3adant3 1132	. . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (𝑈‘(0_g‘𝑌)) = (0_g‘𝐴))
36	35	ad2antrr 726	. 2 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠))) → (𝑈‘(0_g‘𝑌)) = (0_g‘𝐴))
37	1, 10, 20, 23, 24, 26, 28, 30, 36	fsuppcor 9313	1 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠))) → (𝑈 ∘ 𝐺) finSupp (0_g‘𝐴))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1086 = wceq 1540 ∈ wcel 2109 Vcvv 3438 ifcif 4478 class class class wbr 5095 ↦ cmpt 5176 ∘ ccom 5627 ⟶wf 6482 ‘cfv 6486 (class class class)co 7353 ↑_m cmap 8760 Fincfn 8879 finSupp cfsupp 9270 0cc0 11028 1c1 11029 + caddc 11031 < clt 11168 − cmin 11366 ℕcn 12147 ℕ₀cn0 12403 ...cfz 13429 Basecbs 17139 ._rcmulr 17181 ·_𝑠 cvsca 17184 0_gc0g 17362 -_gcsg 18833 ._gcmg 18965 mulGrpcmgp 20044 1_rcur 20085 Ringcrg 20137 CRingccrg 20138 var₁cv1 22077 Poly₁cpl1 22078 Mat cmat 22311 maAdju cmadu 22536 ConstPolyMat ccpmat 22607 matToPolyMat cmat2pmat 22608 cPolyMatToMat ccpmat2mat 22609

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5221 ax-sep 5238 ax-nul 5248 ax-pow 5307 ax-pr 5374 ax-un 7675 ax-cnex 11084 ax-resscn 11085 ax-1cn 11086 ax-icn 11087 ax-addcl 11088 ax-addrcl 11089 ax-mulcl 11090 ax-mulrcl 11091 ax-mulcom 11092 ax-addass 11093 ax-mulass 11094 ax-distr 11095 ax-i2m1 11096 ax-1ne0 11097 ax-1rid 11098 ax-rnegex 11099 ax-rrecex 11100 ax-cnre 11101 ax-pre-lttri 11102 ax-pre-lttrn 11103 ax-pre-ltadd 11104 ax-pre-mulgt0 11105

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3345 df-reu 3346 df-rab 3397 df-v 3440 df-sbc 3745 df-csb 3854 df-dif 3908 df-un 3910 df-in 3912 df-ss 3922 df-pss 3925 df-nul 4287 df-if 4479 df-pw 4555 df-sn 4580 df-pr 4582 df-tp 4584 df-op 4586 df-ot 4588 df-uni 4862 df-int 4900 df-iun 4946 df-iin 4947 df-br 5096 df-opab 5158 df-mpt 5177 df-tr 5203 df-id 5518 df-eprel 5523 df-po 5531 df-so 5532 df-fr 5576 df-se 5577 df-we 5578 df-xp 5629 df-rel 5630 df-cnv 5631 df-co 5632 df-dm 5633 df-rn 5634 df-res 5635 df-ima 5636 df-pred 6253 df-ord 6314 df-on 6315 df-lim 6316 df-suc 6317 df-iota 6442 df-fun 6488 df-fn 6489 df-f 6490 df-f1 6491 df-fo 6492 df-f1o 6493 df-fv 6494 df-isom 6495 df-riota 7310 df-ov 7356 df-oprab 7357 df-mpo 7358 df-of 7617 df-ofr 7618 df-om 7807 df-1st 7931 df-2nd 7932 df-supp 8101 df-frecs 8221 df-wrecs 8252 df-recs 8301 df-rdg 8339 df-1o 8395 df-2o 8396 df-er 8632 df-map 8762 df-pm 8763 df-ixp 8832 df-en 8880 df-dom 8881 df-sdom 8882 df-fin 8883 df-fsupp 9271 df-sup 9351 df-oi 9421 df-card 9854 df-pnf 11170 df-mnf 11171 df-xr 11172 df-ltxr 11173 df-le 11174 df-sub 11368 df-neg 11369 df-nn 12148 df-2 12210 df-3 12211 df-4 12212 df-5 12213 df-6 12214 df-7 12215 df-8 12216 df-9 12217 df-n0 12404 df-z 12491 df-dec 12611 df-uz 12755 df-fz 13430 df-fzo 13577 df-seq 13928 df-hash 14257 df-struct 17077 df-sets 17094 df-slot 17112 df-ndx 17124 df-base 17140 df-ress 17161 df-plusg 17193 df-mulr 17194 df-sca 17196 df-vsca 17197 df-ip 17198 df-tset 17199 df-ple 17200 df-ds 17202 df-hom 17204 df-cco 17205 df-0g 17364 df-gsum 17365 df-prds 17370 df-pws 17372 df-mre 17507 df-mrc 17508 df-acs 17510 df-mgm 18533 df-sgrp 18612 df-mnd 18628 df-mhm 18676 df-submnd 18677 df-grp 18834 df-minusg 18835 df-sbg 18836 df-mulg 18966 df-subg 19021 df-ghm 19111 df-cntz 19215 df-cmn 19680 df-abl 19681 df-mgp 20045 df-rng 20057 df-ur 20086 df-srg 20091 df-ring 20139 df-cring 20140 df-subrng 20450 df-subrg 20474 df-lmod 20784 df-lss 20854 df-sra 21096 df-rgmod 21097 df-dsmm 21658 df-frlm 21673 df-ascl 21781 df-psr 21835 df-mvr 21836 df-mpl 21837 df-opsr 21839 df-psr1 22081 df-vr1 22082 df-ply1 22083 df-coe1 22084 df-mamu 22295 df-mat 22312 df-cpmat 22610 df-mat2pmat 22611 df-cpmat2mat 22612

This theorem is referenced by: cpmadumatpoly 22787 chcoeffeqlem 22789

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